is equivalent to \(S\)  being a Hilbert-Schmidt operator. Furthermore, there is an ample supply of reproducing kernel Hilbert spaces and \(L^2\)  spaces for which the identical embedding is compact, but has an infinite trace, i.e. the embedding is compact, even though the condition (REF ) is violated. For details on those and related issues we refer to [1]}.
In Theorem REF  we give an if-and-only-if characterisation of the compactness of \(S\)  in terms of the reproducing kernel \(k\) .
